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CORNELL NOTES Name _______________________ SOL: Date ____________ Standard A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a)determining whether a relation is a function; b)domain and range; c)zeros of a function; d)x- and y-intercepts; TOPIC: Plotting points and drawing lines on a coordinate plane. Content and Language Objectives Day 1: Create a coordinate plane and label its parts. Plot points in the correct (x,y) form. Connect points to draw a line. Identify the quadrants in a coordinate plane. Day 2: Define relation. Define function. Graph a linear equation. Recognize the domain and the range from a function table, a map, coordinate points, and a graph accordingly. Day 3: Identify x– and y– intercepts of a graphed function. Find the zeros of a function. Vocabulary two-dimensional What is a coordinate plane? plot The coordinate plane is a ____________________ surface on which we can___________ points, lines and curves. How many right angles does a coordinate plane have? ______________ Draw: a Point a Line a Curve x-axis y-axis It has two scales, called the _____________ and _____________, at right angles to each other. The plural of axis is 'axes'. Draw angles: Points on the plane are located using two numbers - the x and y coordinates. These are the horizontal and vertical distances of the point from a specific location called the origin. The center (0,0) acute angle obtuse angle straight angle right angle Origin Where is the ORIGIN? ___________ Ordered pair for ORIGIN: _______ X axis horizontal scale The _________________________ is called the x-axis. As you go to the right on the scale from zero, the values are positive and get larger. As you go to the left from zero, they get more and more negative. vertical scale Y axis The ________________________ is called the y-axis. As you go up from zero the numbers are increasing in a positive direction. As you go down from zero they get more and more negative. Ordered pairs are written in this form: (X,Y) Quadrants The two axes divide the plane into four areas called quadrants ___________________. The first quadrant is the top right, and then they go around counter-clockwise. They are sometimes labeled with Roman numerals: I, II, III and IV. counter-clockwise: (3,2) REMEMBER The coordinates of the origin are (0, 0). The first number in an ordered pair is X Y the x-coordinate. The second number is the y-coordinate. The numbers in an ordered pair are called coordinates. In which quadrant can you find the following points? (4, 6) _____________ (-4, 6)_____________ (-4, -6)____________ (4, -6)_____________ Functions rule A function is a ___________ that associates one and only one ______________ of one value variable with each value of another variable. The function y = 2x expresses y in terms of x. For each value of x, there is one and only one value of y. An equation determines y as a function of x, if for each x, the equation can be solved lines to give exactly one value of y. linear equation The graphs of such equations are________________ in the plane. An equation that can be written Ax + By = C, where A, B, and C are fixed numbers, is called a __________________________. table of values The graph of a linear equation is a straight line. To graph a linear equation, first a _______________________________ for x and y is completed and then the ordered pairs are graphed. Finally a line is drawn through the points. FUNCTION? ___________ Equation: y = x + 3 This is a _________________________________ These values can be interpreted as the x-coordinates and y-coordinates of points in the coordinate plane. Graphing these points means placing a dot at the points (–2, +1), (–1, +2), (0, +3), (+1, +4), and (+2, +5). To graph the equation, students connect the points with a straight line. FUNCTION -OR- RELATION What is a relation?____________________________________________ What is a function?____________________________________________ An connection) of members of the domain (x) with exactly ONE MAPPING Example: A. Example: B. 9 12 6 0 21 5 6 0 8 12 21 10 0 3 6 6 1 Ex. A. is a ___________________ Ex. B. is a ___________________ Why? _________________________________________________________Example A has each “x” associated (co )CO-ORDINATE POINTS Use the mapping above to make co-ordinate points: ( Ex. A. ( , ) , ) ( ( , ) ( , , ) ) Ex. A. is a ___________________ Ex. B. ( , ( ) , ( ) , ( ) ( , , ) ) Ex. B. is a ___________________ Why?______________________________________________________________ In Example A no x is repeated. In Example B the “x” repeats BUT the “y” GRAPHING 4 4 3 3 2 2 1 1 Ex. A. Ex. B. -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 -1 -1 -2 -2 -3 -3 -4 -4 Ex. A. is a ___________________ 2 3 4 Ex. B. is a ___________________ Why? _______________________________________________________________values. This fails t DOMAIN & RANGE X values What is Domain? ___________________________________________________ What is Range? ____________________________________________________ Y values input _____________Example A no x is repeated. In Example B the “x” repeats BUT the “y” 4 4 for each is d output -4 -3 -2 D: { R: { 3 3 2 2 1 1 -1 1 2 3 4 -4 -3 -2 -1 1 -1 -1 -2 -2 -3 -3 -4 -4 } } D: { R: { 2 3 4 } } _______________________________________________________________________ 9 12 0 6 12 21 5 21 6 0 10 0 8 6 3 6 1 D: { R: { } } D: { R: { } } _________________________________________________________________ Again, use the above mapping to make your coordinate points. ( , )( ( , , )( , ) ( , ) ) ( , ) ( ( , , ) ( ) ( , , ) ) IDENTIFY THE x – and y – INTERCEPTS from GRAPHED Functions What is intercept? _________________ x– intercept _______ x– intercept ______ x– intercept ______ y– intercept _______ y– intercept ______ y– intercept ______ FIND the ZEROS of a FUNCTION The ZEROS of a function are the same thing as the _______________________. 1. 2. zeros: ________________ 3. zeros: ____________ 4. zeros: ________________ 5. zeros: ________________ zeros: ____________ 6. zeros: ____________ 1.